Problem 25 - Chemistry Matter and Change

Question

Challenge A flexible plastic container contains 0.860 g of helium gas in a volume of 19.2 L. If 0.205 g of helium is removed at constant pressure and temperature, what will be the new volume?

Step-by-Step Solution

Answer

The new volume of the helium gas will be approximately 14.62 L.

Step 1: Understand the initial conditions (ideal gas law)

According to the ideal gas law, at constant pressure and temperature, the volume of a gas is directly proportional to the number of moles. Initially, the container holds 0.860 g of helium. The initial volume is given as 19.2 L.

Step 2: Calculate the number of moles initially

To find the initial number of moles of helium, use the molar mass of helium, which is approximately 4.00 g/mol. Calculate it as follows: \( n_i = \frac{0.860 \text{ g}}{4.00 \text{ g/mol}} \approx 0.215 \text{ moles} \).

Step 3: Calculate the number of moles removed

Calculate the moles of helium removed using the mass removed (0.205 g) and the molar mass: \( n_r = \frac{0.205 \text{ g}}{4.00 \text{ g/mol}} \approx 0.05125 \text{ moles} \).

Step 4: Calculate the new number of moles

Subtract the moles removed from the initial moles to find the new number of moles: \( n_f = 0.215 - 0.05125 = 0.16375 \text{ moles} \).

Step 5: Apply the ideal gas law to find the new volume

Since the pressure and temperature remain constant, use the proportion: \( \frac{V_i}{n_i} = \frac{V_f}{n_f} \). Substitute the known values: \( \frac{19.2 \text{ L}}{0.215} = \frac{V_f}{0.16375} \). Solve for \( V_f \): \( V_f = \frac{19.2 \times 0.16375}{0.215} \approx 14.62 \text{ L} \).

Key Concepts

Moles of Gas

In chemistry, a mole is like a count of items, similar to how a dozen means 12. One mole of any substance contains exactly Avogadro's number of particles, roughly \( 6.022 \times 10^{23} \) particles. For gases, moles can be determined using the molar mass, which is the mass of a given substance divided by its molar mass. In our problem, we initially had 0.860 grams of helium. The molar mass of helium is approximately 4.00 g/mol. So, we calculate the moles of helium initially present as:

• \( n_i = \frac{0.860 \text{ g}}{4.00 \text{ g/mol}} \approx 0.215 \text{ moles} \)

Understanding moles is crucial because it helps us connect the mass of a gas to other physical quantities, like volume and pressure, using gas laws such as the ideal gas law.

Volume Change

When some helium gas is removed from the container, the volume of the gas must change if the pressure and temperature are kept constant. We start with a volume of 19.2 L and after removing 0.205 g of helium, or \( 0.05125 \text{ moles} \), the total moles decrease. The volume then changes accordingly. To calculate the new volume, we use the proportional relationship between volume and the number of moles:

• If initially we have 0.215 moles for 19.2 L, then after removal of gas, we have \( 0.16375 \text{ moles} \).
• The new volume \( V_f \) can be found using the formula: \( V_f = \frac{19.2 \times 0.16375}{0.215} \approx 14.62 \text{ L} \).

The new volume is thus 14.62 L. This demonstrates how volume and quantity of gas are directly related when they are isolated from other changing factors like pressure or temperature.

Gas Laws

Gas laws describe the behavior of gases, typically under various conditions of pressure, volume, and temperature. There are several gas laws, but for this exercise, the ideal gas law is key. It formulates the relationship between pressure, volume, temperature, and number of moles of a gas:\[ PV = nRT \]Where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

In practice, for constant pressure and temperature, the ideal gas law simplifies to a direct proportion between moles and volume:\[ \frac{V_i}{n_i} = \frac{V_f}{n_f} \]This relation lets us predict the effect on volume when the amount of gas changes, assuming other conditions remain constant.

Molar Mass Calculation

To determine the number of moles of a substance like helium, calculating its molar mass is necessary. Molar mass links a material’s mass to its number of moles. Helium's molar mass is about 4.00 g/mol. To calculate moles, you divide the mass of the gas by its molar mass.

• For 0.860 g of helium: \( n_i = \frac{0.860 \text{ g}}{4.00 \text{ g/mol}} = 0.215 \text{ moles} \)
• For 0.205 g of helium (removed): \( n_r = \frac{0.205 \text{ g}}{4.00 \text{ g/mol}} = 0.05125 \text{ moles} \)

This step is foundational for solving problems involving gases as it provides the link from mass, which is easily measured, to moles, which relate directly to gas volume under constant conditions.